PTFE will not produce fumes unless it gets to above about 500°F, and once it all burns off it's done producing fumes. That's the good news. The bad news is that a toaster oven is definitely something that could contain some. And the heating elements themselves could easily get to those temperatures.
Odds are high that the heating elements have residual PTFE from the manufacturing process. I would assume this is the case. If the toaster oven is not new and has been used many times previously, this is not likely to pose a danger. Any new appliance with a heater should be "burned in" outside for a period of time.
It is also possible that any nonstick coating on the lining of the toaster oven contains PTFE. You could contact the manufacturer and they should be able to tell you this if you give them the model number. Any baking pan that came with it could also have a PTFE-nonstick coating. It could alternatively have a silicone coating which is not harmful.
Depending on how you use the toaster oven, any PTFE in the lining might or might not ever have a chance to get hot enough to pose a risk. If you broil a lot or bake on the highest heat settings, it is possible hot spots could form. Hot spots are the main reason people have the idea that PTFE can create fumes at lower than 500°F. It cannot, but a largely empty skillet on a gas burner can exceed that temperature at the edges even if the food in it, and the pan underneath the food, is at a "normal" cooking temperature.
PTFE fumes kill birds almost instantaneously. They actually cause flulike symptoms in humans as well, which makes me wonder how it is legal to use the material in self-cleaning oven linings, but I digress. If your bird is fine, you oven is very likely safe. But you may want to contact the manufacturer anyway.
As for self-cleaning ovens, even if they don't contain PTFE or the residual has burned off, the gunk from burning off the organics is probably not optimal. I take my birds outside until the cycle has run and the air is well cleared out. And that was with an oven that had run through the cycle a number of times. The normal gunk is a ventilation issue as far as I know, in a large well-ventilated house where the odor doesn't reach the birds you are probably OK. But PTFE fumes are a whole different deal. They spreads rapidly and a very small amount is fatal. We recently moved and have an oven that I do not know if it was ever used on the self-clean cycle. I don't know if I'll ever run it. It would have to be on a day when the birds and I could be outside all day and the house could be completely opened up to air out and I'd still be really nervous. As in I'd consider bringing a sacrificial finch I wasn't attached to home first!
I recently got a new toaster oven. I burned it in on the porch through three times the recommended cycle (the instructions actually specified a burn in period). All has been well. I will admit to some trepidation upon initial usage, but if I'm going to argue logic, I need to put my money where my mouth is and believe in what I know!