Do you need to have cars there or just a workspace?

These would have to be professionally done.

Both. My wife stores her mid size suv there. Currently my truck does not fit. I'd also like the option of adding a slightly taller garage to fit my vehicle. The room for work is much needed as tools take up space. Also, it's where my large dogs hang out when we're away and it's hot out.

They can all be moved, it's only a matter of money. It's probably enough money (guessing a couple thousand per service, depending on exact circumstances) that I'd look carefully at alternative design strategies.

A main electrical panel with many incoming wires can be quite a challenge to move far, as evey splice will need to be in an accessible box of the appropriate size. An electrician can advise you of what exactly is involved. Plumbing and gas is usually easier because fewer things need to be re-routed.

You need to ask an electrician for sure and spliacing wires is not IMO a safe option so lots of work involved to move a main electrical bow . Why not post some pics of the plan and of the house to see if there might be a better option . IMO 4 car garages often look very out of proportion to a home .Maybe what you need is a separate building for the truck and workshop.

A new SER cable would need to be run from the main panel to reach the new locaction. There would also need to be a bunch of juction boxes to extrend all the home runs, or all home runs would need to be pulled again.

Keep the utilities location and design the addition with that in mind. This is an expensive enough of a project without adding another 10K here and there because it doesn't actually work with the existing home’s architecture and infrastructure.

Those utiity locations are defined as an "existing constraint." Design around the service entrances. I can't imagine you gain anything in the layout of the garages that can justify spending that much money on moving them. Consider the costs as new services, they will retap the water, probably most of the gas line will need rerouted (because can be spliced if newer PE), and a new power feed from the pole + distribution panel and making the existing a sub. Doesn't even get into if you need to bump up 100A to 200A or 400A.

"You need to ask an electrician for sure and spliacing wires is not IMO a safe option so lots of work involved to move a main electrical bow ."

Every house already contains many hundreds of wire connections (aka splices), both through devices and directly. Done with the proper connectors inside an appropriate enclosure why do you think they are not safe? I don't think there's any evidence or reason to support that.

I've relocated both gas meters and electrical boxes and, depending on exact circumstances, it doesn't need to be that big a deal. If it greatly improves the design outcome it can easily be worth the couple thousand dollars it costs. That said, I'd certainly think through the alternatives first.

Get quotes on the cost of those relocations before you commit to a final design. I agree with the view that this could be a pricy additional cost, and the money could be better spent elsewhere on this project. I've done several remodels where I made the design accommodate the existing utility locations rather than go through the hassle and expense of moving them. Getting an architect to look at this might give you a better design and avoid unnecessary costs.

Is there another option to adding the two car garage to the exiting one? Explore those options unless you're prepared to hire pros to prepare to move the utility meters.

For the gas company and electric company and water company to move their meter, you'll first need to have a pro create the new place for the meter to be moved ... and those utility companies will require the new bases for the meters be up to their standards and/or code.

You never posted a floor plan, but I'm assuming your panels, etc. are on the side of the garage away from your house. If you expand the garage in that direction, could you put the panels between the garage doors?

The garage is currently a tandem 3 car garage. My wish is to convert the 3rd tandem garage into a den room, and add that additional two car. Offsetting the garage would make it very awkward as it would be approximately 6-7 feet, and making it into an outside shed would be useless. I was quoted about $3-6k for the electrical panel move. Where I live we can't have the panels inside.

For context the red boxed area is where all electric panel, gas and water metering are, as well as where they'd be relocated to.

Just some thoughts:

If the red is where the meters are and the green is where you want to move them, wouldn't the meters still be inside the new build if it is the rectangle outlined in green in the upper left or do you mean the red is where the meters are now and you want to move them straight across in the direction of the arrow to the outside of the green rectangle ... w/green rectangle representing the new build?

In your plan, to the right of where there are the words "7' side yard setback" , there appears to be a part of the garage that is narrower (with angles) ... perhaps used for mower, tools, etc ? -- but still a part of the garage. IF you create your new build beside that back part of the existing garage -- behind the red marks -- would your new build be behind the meters and your meters still outside? How far from the building do the meters extend? Would they be in the way of driving a vehicle past them to park within a new build built behind them -- behind where the new build is now shown? For a larger car, perhaps, but would a smaller car be able to drive past the meters to reach a new build farther back behind the meters?

Check local codes and see if having the meters on the side exterior wall of the garage but beneath an open car shed/porch still considered "inside" and prohibited; if not, consider adding a car shed rather than adding an enclosed garage.

There appears to be two "cable tv" easements -- one in front and one in the back right.

Are there any meters actually behind the house where there is a "cable tv utility easement" on the bottom right behind the house? Have you considered/priced moving the right rear cable tv access point and creating a car shed/porch at that back right corner of the house rather than adding to the garage in front?

the green room represents the would be added two car garage. It's currently a tandem 3 car, but I would cut the third garage, turn it into an office/den, and re-expand to the side. The meters extend at most 2 feet away from the wall. The meters would be relocated directly to the other side as the arrow projects- but on the outside wall of course. With this I would have about 8 feet of setback/space from the new garage to the end of my property.

Pushing the new two car garage back into the backyard is counterproductive for what I want. For starters, the push would be substantial, about 7 feet. On the inside, I would like the space wide open. An offset that large would disrupt the space. I may have mentioned before, but this large garage is meant to serve various purposes. Not only would it house my wife's mid size SUV (Kia Telluride), but also have cabinetry for wood working, a dog shower station, some basic gym equipment like a treadmill, squat rack, finally fit my truck, and still have enough space for my 4 100+ pound dogs to sleep in and hide from the heat when I'm away.

With all that in mind, I'll check look into permits and keep referring with people with experience to really know the price ball park. Being that it's a pretty big project, a few thousand is a smaller price to pay for what I'd have in the end.

Buy a different house.