Maspalomas is the “standard” place to stay, there’s thousands of roadies head there and it’s probably the main tourist town on the island after Las Palmas. Lots of bars, restaurants, big sandy beaches and so on so it might fail the “quiet” part of your requirements. I’ve only been to Maspalomas once for half a day cos I needed to go to FreeMotion, the bike hire place so I got the bus there. Didn’t like it much but I didn’t really see all of it. It was noticeably warmer than the northern half of the island though!
We’ve stayed in Teror, a sort of big village/small town up in the central north and that’s lovely. Couple of decent supermarkets, enough bars and restaurants to get by on (not a massive choice it has to be said) but it’s quiet, it’s easy access to all the decent road riding; Maspalomas comes with a long drag out and back every single ride to access the best bits like Valley of the Tears and Pico de las Nieves.
They had some really bad wildfires there a couple of months ago (predominantly in the north of the island, around Tejeda), worth checking if any roads are still closed before you book accommodation. You don’t want to be stuck on only one road out and back!
It’s an amazing place, stunning road riding. Wider tyres are a good idea though – the roads tend to be either beautifully smooth or “like British roads”, there’s not really an in between.