Did qui gon know padme was the queen?

Yes but it was probably Qui-Gon AND Obi-Wan's job to protect her. I'm sure she'd be much safer with Obi-Wan in the ship than with Qui-Gon at Mos Espa.

I agree with that. Qui-Gon had never been to Tatooine before so he had know idea what to expect. She would have been much safer on the ship.

So I think we can safely presume that he certainly didn't know it was her when they first spoke in the desert by the ship. So why would he know it was her anytime later?
I don't think he ever knows at all. She puts on a great act, so that even a Jedi can't suss her out.

And Qui-Gon had more important things on his mind.

Yes, in fact, he seems to totally ignore Padmé most of the time.

That is a good point master Daft.

I think he did not know in the begging...(at the ship w/panaka), but it did seam like he started to sense it later and was kind of baiter her or teasing her once and a while...

By the time he may have realized it, it was too late to bring her back to the ship and thought he would keep a close eye on her...

On the other hand....he did leave her too much ...I'm still not sure, but did not look particularly surprised like the others at the end...he was more intersted in ob1's reaction....sort of..

Yes I guess it's very possible that he knew, especially with that sort of smile on his face when she reveals it. He looks like he wants to say, "Ha, I knew it. I just knew it."

Obi-Wan smiled as well but he hardly knew her.

Good point. Well to be honest I still don't think Qui-Gon knew anyway. I was just saying ideas that could explain why he might have.

I think they both knew..what kind of jedi would they be if they didn't?

I think Jedi just sort of sense emotion and feelings of others. Like when Qui-Gon says "I sense a great amount of fear" and when Luke senses his friends' pain while he's on Dagobah.
Maybe if Amidala had have been nervous about her disguise then Qui-Gon would have sensed it was her, but she managed to stay calm.