You can frame a floor opening, yes. However, your plan to brace the cut joist is not appropriate.

https://www.jlconline.com/how-to/framing/field-guide-to-common-framing-errors_o

Basically, the joist that you want to cut will be mounted to the header joists, which will be mounted to the adjacent, uncut joists.

This is something you definitely want to permit and get inspected.