It's not true. Without wishing to condone or play down the dictatorial nature of Qatar's ruling family, the place is a positive hotbed of liberalism and equality compared to the rest of the region. There are not lots of poor unemployed young men in Qatar, who have been most active throughout the rest of the Arab world. Furthermore, Qatar is home to Al Jazeera, who are the voice of the Arab revolution, and having AJ in Doha has made the Arab populace very sympathetic to Qatar's ruling family. What's more, most of Qatar's population are imported workers from the Indian subcontinent, and have no interest in political change in Qatar.
Yes, they've locked up a dissident blogger. That is an absolute disgrace. Tragically, it is not all that uncommon. It is not a signal that the place is about to go into meltdown. The chance of the Qatar GP being called off are minuscule at best.
Now, Saudi Arabia, that's a different question altogether.