Is the gas fireplace vented to the outside, or is it a ventless type??

If this is a combo alarm, or just a CO alarm - you need to immediately have the fireplace repaired by a qualified individual in order to be able to use it.

If this is just a smoke alarm (smoke alarms are particulate based / fire alarms are heat based - (rapid temperature increase)) - you still have some sort of problem going on.

A ventless fireplace has an low oxygen sensor on it; but you still have to be aware of CO and combustion byproducts. A ventless unit - ALL compbustion byproducts and generated heat are deposited into the room.

A vented unit should have an outside vent; and room air, and combustion air should be totally separate. Combustion air should not be in the room at all, in any way shape or form - with a vented unit. These function somewhat like a small furnace.

Ventless units are supposed to be extremely efficient - which is why they can be 'ventless'. It is a good idea, on ventless - to have a window open slightly when operating.

You DO have a problem with the fireplace - since the 'smoke' alarm is going off when you turn the GAS fireplace on... A wood burning fireplace can have chimney height issues, draft issues, etc, which could cause a smoke alarm to go off - with an actual wood burning fireplace - a smoke alarm going off - would not necessarily be an actual true health safety problem. But would indicate that the chimney has some sort of draft problem. (Plenty of wood burning fireplaces do have draft problems.)

A GAS fireplace does not have these type of problems. And the functioning of this gas fireplace needs to be repaired by a qualified individual asap.

A gas fireplace burner installed into a wood burning fireplace box - could be not burning cleanly - and having particulates going into the room... sort of like a wood burning unit.

If this is a gas fireplace vented unit; and not a ventless or a wood burning converted unit - you could check, and see if a vent is clogged by a birds nest or something. The sealing of these things is not watertight. They work somewhat on the principle of "least resistance". (combustion air is kept separate by pipes; but these pipes are mostly just tight fitted together - they don't have a bunch of sealant. And a burning fire is looking for air to burn...) So if a vent is visibily clogged; your clearing of the vent might easily solve your problem. But if you don't see anything obviously wrong when looking at a vent - get it repaired. (or don't use.)

Mostly - it sounds like this needs some sort of repair, and concerns your saftey and health.

And has NO SMELL. You won't SMELL it. It is COMPLETELY INVISIBLE to ALL of your senses.

Your red blood cells will bond to it more easily than oxygen. And it can kill you quite easily. Poison you very slowly.

If your fire alarm is a combo unit (smoke and CO) that is a great saftey feature for you; and again - your fireplace would need work like what I suggested above.

If you're in a traditional house, your bedrooms are upstairs. If a fire starts downstairs (fireplace or otherwise) I'd rather be altered to that fire as quickly as possible, not after it's already going so much that the smoke reaches my bedroom upstairs and I have to jump out a window instead of going downstairs.

To the OP-yes, check on the CO. But either way, have the fireplace checked by a chimneysweep to make sure it's venting properly. It could also be putting enough particulates into the air to cause this. It's not right, no matter what.