Squeak is right in that in real life, qualifying offers are not submitted until the end of the year. We've never really clarified any kind of time frame in our game for qualifying offers in our game, which is something that probably should be adjusted for next year.
The way I always have looked at it, a qualifying offer is any offer that meets the criteria under Section 4.2.1 of the CBA.
Note that while there are no limits on the number of qualifying offers you can submit, any full negotiations or preliminary discussions still count towards the limits discussed in section 4.5 of the CBA. Agents should not respond with any sort of counter offer to a qualifying offer unless the team indicates it wishes to hold full negotiations or preliminary discussions with that player.