You should take the GRE general exam anytime before the Fulbright’s application deadline. You can even take the GRE on the day of the deadline.
For 2020, the Fulbright deadline is 15th June 2020.
Do note that once you take the GRE, you have to send the scores from the ETS directly to the USEFP and you can do that by entering the code 9388 while sending your official scores to USEFP. If you plan to do this on your GRE test day, you can do that for free. If you plan to do it after your GRE test day, then you will have to enter the same code but pay an additional sum to send the official scorecard to USEFP.
Your application to Fulbright will be discarded if you take the GRE after 15th June 2020.