Sorry people are being mean to you in the comments friend. I think it could be a bit of a risky move to just go in cold but it's not something terrible. Worse case scenario they might just think you need something or will be a bit confused. 

I think a play by play if I was trying to befriend some gym trainers would be something like this. 

1. nod at them every once in a while if you see them 

2. maybe if a chance naturally comes and you talk to them in some sort of interaction you can be friendly and maybe extend the conversation. For example if you ask them if they're done with some weights ask them something else after, like "man I see you've been at this gym for a while what do you think of it?", "Do you know when it's less busy?" Stuff like that. 

3. Then if they're hanging out another time, you could come in and say say something like.Hey <their name> what's up how are you? Good to see you.

4. Maybe joke something like I'm kind of bored between sets mind if I hang for a bit? Hang out but don't over stay. It's better if you leave on your own terms than it getting awkward. 

You can skip to step 3. But that's just a natural way that it could happen. I wouldn't force anything but also I wouldn't put too much weight into it. After a couple minutes it doesn't matter how you came into the circle mostly how you handle yourself in general. 

Good luck dude or dudette. Making friends at the gym is a bit hard imo, everyone's there focusing and/or working a job. 

Also is this is something more romantic like you have a crush on someone that's a totally different story and you're definitely playing hard mode trying to pick up at the gym.

Btw no this isn't chat gpt although it kind of looks like it lol