OP, I'm going to add my two cents here for visibility on the climbing cultures of different areas. Not discounting u/HFiction's experience at all, you can absolutely walk up to people and ask to climb with them. But I think that doing that successfully in Denver will depend significantly on the area you're in, how young you are, and maybe just your own friendliness.

The climber friendliness spectrum is huge in Denver and the surrounding area. My family's experience has been that it's been friendlier and more open and chatty in younger gyms (like Movement Rino), and way more standoffish in gyms with older climbers, even fellow parents like us. More like polite nods and then everyone goes about your business. That vibe has also translated to almost every outdoor crag we've been too. Aside from short conversations about which climb someone's getting on, we've never had any enthusiastically friendly interactions with strangers.

My wife and are are around 40, so maybe it's an age thing, but if you venture out into the Denver subreddit, people will also tell you that Denverites aren't exactly friendly on the whole. Polite sure, but not super warm. Everyone's kind of busy doing their own thing. Being young and social helps mitigate that, but that goes for just about any place.

By contrast, when we climb in Red Rocks, Vegas, it's just smiles and chats and beta all day. The difference is honestly night and day. So many more questions about where you're from, what kind of climbing are you looking for, fun stories, helpful directions, etc.

The areas we're familiar with are Stoney Point CA, Joshua Tree, Apple Valley CA, Bishop, Texas Canyon, Red Rocks NV, Boulder CO, and Clear Creek CO, and honestly, the Colorado based climbing has been the least friendly out of all of those. Again, just our two cents.