I was searching for the same a few weeks ago! Having recently landed in Taipei, enchanted by this country, and inspired to try my first cycling tour in a place I feel safe and happy to ride, this is what I found re: buying second hand bikes in Taipei:
I ultimately bought a second-hand bike from Mathew and May who run MathewBike, a local bike shop servicing around-the-island tours. Transparent pricing table on the site for rentals. Since my trip is open-ended and i didnt want to be constrained by a return date, I asked about buying a bike. Very small selection as its a small storefront, but everything in there is equipped to handle an island tour. I got a primavera road bike, all equipment included (rack, panniers, lock, helmet, lights) for $6500. Pretty close to their month-long rental price, so I feel happy with the price and I imagine I could sell it back to them or elsewhere (or just give it to them 'cause I’ve come to appreciate them so much!) Mathew tuned up and cleaned the bike, ensuring its fit for the journey ahead. Both of them are knowledgable about preparing bikes for this specific type of adventure. This, plus coming to befriend the pair and see them as wonderful souls furthering the bike scene in Taipei, and the bike being close enough to what I wanted, made me happy to buy from and support them. I plan to start cycling next week after some camping supplies arrive in the mail
Facebook Group: Taiwan Bike Market
Pros: Some quality bikes and parts
Cons: Not geo specific to Taipei, listings from all Taiwan. Needle in a hay to get the specific build I want, especially because I am short and need a less common frame. I used the search function and made a post. Didnt find anything that fit me, YMMV.
I went to some local bike shops whenever I passed by them whilst walking around the city. Most were small and didnt have something in my size, much less secondhand, so nothing I’d recommend here.
Giant
At the Liv flagship store, which didnt carry used bikes, I was directed to another Giant location which may have secondhand bikes. I never went, as I found a bike I was happy with before then, though this could be another resource for someone:
No. 153-155, Section 1, Sichuan Road, Banqiao District, New Taipei City, 220