I *just* heard this song (Spotify? New Music Friday?)
I'm going to disagree with the other two posters: "Pushing 40" can mean that you are nearly, but not quite, 40 years old.
However, in this case, I think they're talking about speed and the 40 is 40 miles per hour (and pushing 40 would mean going almost, but not quite, 40 miles/hour - somewhere in the high 30s.)
Metaphorically, it makes more sense. The whole verse is about movement. He's describing trying to get out of the "friend zone," but he's going too slow to do so.
Also, this particular singer is already over 40 years old. He's done pushing 40. He pushed it. He's gearing up for 50.