If you are looking for a permanent cable run, rather than those fuzzy-brush covers (which have zero insulation or fire-block potential) you can get in-wall ethernet wall plate ports that will seal air out and provide a permanent ethernet receptacle in the wall. The advantage of these is that once you install them, you can disconnect the cable from either side of the wall if necessary, rather than having to snake the cable back out of the the wall again. In your case, even if you're just going from one side of the wall to the other, this is worth doing if the cable would stay there permanently or if you plan to pipe Ethernet in even after moving the modem to the garage later (not that I recommend that - just keep it inside if you can).

https://www.homedepot.com/p/Commercial-Electric-Category-6-Jack-in-White-10-Pack-5016-WH-10PK/305193884

https://www.homedepot.com/p/Commercial-Electric-White-1-Gang-1-Decorator-Rocker-1-Duplex-Wall-Plate-1-Pack-5001-WH/206428112

If you use foam to fill any voids or gaps around the cable, make sure you use fire-blocking foam inside the wall (can get it at any hw store). It's required by code in some areas, better safe than sorry. This is for inside the wall, i.e. you don't want to see it because it's typically a bright orange color.

I recommend you use a shielded and outdoor-rated Ethernet cable if possible to cut down on interference. Most good quality cables now are shielded, but wanted to mention just in case.

the smaller the hole, the smaller the heat transfer. you can also fill it with some kind of montage foam, once the cables are through.

Passing cables through existing walls nearly always involves taking part of the drywall off to gain access to the core of the wall. If you need to feed a wire across an entire wall, you typically have to cut an entire strip of drywall off along the entire length of the wall.

If you have access to the beams beneath the floor or in the ceiling, you may want to do most of the run there, then drill into the desired wall through the sill or header plate. That allows you to get the cable into a specific stud gap to limit the amount of drywall affected.

But unless your house was built with cable pipes/runs built into it (and I can’t imagine this being done outside of commercial buildings), you have lots of futzing around to do, and no small amount of drywall work.

https://discuss.online/comment/212555

So I did a detailed write up on this comment about how to run cable.

Doing it in a wall with insulation requires a little more work when fishing with a fish tape, but is totally something a normal person can do.

If the garage is too hot for your router, it probably isn't the best place for your network and/or server gear either. Unless you're intending to water-cool it.

(To put it another way, are you absolutely sure you've thought this through and there is no small-network-rack-sized space in the house? Under the stairs, in the back of a closet, or on top of a cabinet, maybe?)