No. Canada has an agreement with the US that nobody entering Canada from the US will be seen as a refugee and will be deported back to the US. Vice versa is true with people entering the US from Canada.

It's the Safe Third Country Agreement. They even have big red letters saying "yes this agreement is still in effect".

Look into the Rainbow railroad

simply being lgbtq+ isnt eligble enough, you would have to face persecution, as lgbtq+ isnt threatened as a whole in the USA in the near future(jailing, killings, wartorn). eg,some place fleeing from IRAN AS LGBTQ+ would be eligible, or of a specific demographic like religion(ba'hai). You can still flee to safe spaces in blue states, blue areas, so its not really eligible, unless all the states starts targeting lgbtq+(with above persecutions), right now its mostly rhetoric and anti-trans laws here and there, and not really enforced as a whole for the most part.

if you have specialty, in stem try looking for certain positions. though rare, scientist "refugees" are possible.

Last I heard, refugees needed to demonstrate a clear, immediate, concrete, mortal threat, that you couldn't feasibly escape anywhere else in your country.
And if you're rejected you may not reapply and you may even be barred entry.

But I'm not a lawyer, so idk the details or what would qualify as such a danger (it even if I was given the correct information, so take it with a grain of salt).

I suspect that for the most part, probably not.