Should you show more then one payment durations on SaaS checkout pages?

On my SaaS app, I currently only have a recurring monthly option.

Should I also add yearly and quarterly payment options (and offer customers a small discount if they pick one of those)?

One clear benefit of doing so would be getting more instant cash flow. Another would be "locking customers in" for a longer time period and reducing churn.

One possible downside would be that showing three options would clutter the UI and lower conversion rates.

So many other SaaS sites offer multiple durations, so I wouldn't think that you are cutting new ground here (but cannot be sure without knowing your market).

Off the top: Have you asked your (current) customers? Captured those who haven't didn't sign up and ask them? Instrumented your signout page to determine dropoff rates? Have a free trial option, then poll those who didn't sign up?

Only way to know if your conversion rates will be impacted is to do A/B tests with your particular case and see the results.