I learned that "I feel a toast coming on" is equivalent to "I propose a toast".
I'm curious about the grammar of the phrase "I feel a toast coming on".
S1 feel S2 V+ ing equals to S1 feels that S2 is coming?
It's playful use of language—almost a joke. "I feel a toast coming on" echoes the following old and familiar expression:
Notice that the expression primarily pertains to illness. It suggests that the illness is overcoming you against your will.
People have playfully applied the phrase to a number of other things, suggesting, partly as a joke, that their desire to do something is like an illness overtaking them, forcing them to do something against their will:
As with "I feel a headache coming on", these suggest that the person is going to sing their song or make their speech right away: just as if it were a headache, they cannot volitionally prevent the song from occurring. It also playfully suggests, just a little, that needing to give the speech is a sort of illness—something wrong with them, but something that they (and you) must tolerate. With a song, I there's less suggestion that the the listener won't enjoy it, though that element of the analogy can be played up.
"I feel a toast coming on" is just a variation of "I feel a speech coming on."
So, to answer your question: No, "S1 feel S2 V+ ing equals S1 feels that S2 is coming" is not a general grammatical equivalence in English. It only makes sense if you want to draw a parallel with the way you feel the onset of an illness.
However, it is generally true that quirky English grammatical constructions work by drawing parallels with older, familiar expressions, usually adding some new twist or variation to suit the present situation. This is why English grammar does not submit well to being characterized by exact rules, and why it takes so long to learn. You have to get acquainted with the familiar expressions that the variations refer to.
Note that the word "on" is necessary. "Come on" is a phrasal verb: both words are needed. "Come" by itself does not mean the same thing.