It rhymes with "cheer" and "veer".
However, it's important to note that the pronunciation of "queer" has evolved over time and can vary depending on individual preferences and regional dialects.
Some people may pronounce it with a more traditional pronunciation, such as kweer (rhyming with "beer").
Ultimately, the best way to pronounce "queer" is to ask the person who identifies as queer how they prefer it to be pronounced.