Brook is a good fighter but every time he has faced an elite guy, he has been stopped. Crawford, Spence and GGG all stopped him and I think so would Quartey who was one of the strongest welterweights ever. I think Ike breaks him down and stops him in the later rounds.
To be fair to Brook he moved up to Middleweight to face Golovkin which was a very bold move he was out of his best weightclass and suffered a terrible eye injury which effected him for the rest of his career. Quartey for example only had 1 fight at Middleweight and lost in a one sided fight to Wright so I can't see him doing much better vs a prime Golovkin either. I thought Brook was beating Spence for example until the eye injury in the later rounds and vs Crawford he was already considerably past his best. Whilst I do think Quartey would still beat Brook I do think some context is needed for some of the fights you mentioned.
I think Brook would look good for 4-6 rounds using his speed. But, as the fight wore on, Ike's punishing jab would start to wear Brook down until Ike went in for the finish.
My inclusion of GGG was strictly about pointing out his losses. Do I think Quartey would beat GGG? Hell no. Regarding Quartey's losses, the only two legitimate losses in his career were too Vargas and Winky. The ODLH fight could have gone either way ( I personally think IKE won) and the Forrest fight was a joke. At WW, Ike was unbelievably strong and I don't think Brook could handle him.
I know I'm not having a go at you it's just the eye injury Brook suffered vs Golovkin did effect him for the rest of his career and that was a big part of why he lost to Spence who I thought he was beating until the eye problems in the later rounds. Had Brook not fought Golovkin prior to facing Spence that fight could've been a whole lot different although that's assumption on my part. Regardless we both agree Quartey beats Brook I think Brook is a step below Quartey overall and would get stopped in the later rounds IMO.