I don't think Hal and Guy are simply the case of 2 alpha males clashing, or at least, it shouldn't be. Other than Hal, Guy has been there the longest among the Earth lanterns and so their histories really go way back. If I'm not wrong, Guy first appeared when Hal learned that Abin Sur could have either chosen him or Guy but he only got the ring because he was closer (something I'm glad got retconned). Hal then went to meet this Guy person in his civilian identity and they hit it off pretty well.
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In Emerald Dawn II back in the 90s their history was told differently and turned out they first met when Hal got arrested for drunk driving (again, something I wouldn't miss being gone) and Guy was his caseworker. The reason Guy got the ring was because Hal was constantly away from Earth so the Guardians needed a backup. Nobody really liked Guy because he kept picking fights, had anger issues and yeah, was kind of a jerk. They wouldn't think of him as the real Lantern, so he challenged Hal to a fight; the winner got to keep the ring --> he lost.
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There were panels here and there during Hal's tenure where Guy showed up. There was this one time Hal chose a mission somewhere far away in deep space or something to get some me-time and Guy just barged in because he thought Hal would be bored. But there are also a lot of panels in which Hal just ignored Guy or punched him in the face. Unlike modern interpretations, Hal was actually quite stoic and took his work seriously. Guy was the ******* in the League, who picked fight with Bats, constantly got under everyone's skin and made people go "where is Hal?". Guy was the one that was reckless and impulsive, so he got beat up a lot (though in his defense more often than not his bites are actually as loud as his barks). I think like most people, Hal finds Guy annoying and unprofessional, but if speaking as two men I think Hal seemed to like Guy.
There is certainly some bad blood between them however. Guy is known for having the need to prove himself, and the fact that he was usually in Hal's shadow really didn't help. And then there's the thing with Parallax, so realistically speaking, Hal shouldn't be the first person Guy voted for when it comes to being trustworthy. Oh, and don't forget the mess surrounding Kari Limbo. So overall, my take on their relationship is that they are colleagues who grudgingly respect each other, who would go out for a drink together, doing fist bumps but are not so close they would call each other bros.