Google Books has just four legible matches for the continuous verb form...
Time for me to be hitting the [hay / sack / road / ...]
...compared to at least dozens of matches for the plain infinitive form...
Time for me to hit the...
Personally, I think they're just two possible alternative phrasings. I doubt even the most "careful" speaker would consciously choose the less common form specifically in order to convey some subtle nuance of difference. That would be a bit pointless anyway, since there's almost no chance that the audience / reader would understand any such nuance.
Having said that, it's just about possible to suppose the progressive form carries greater urgency / immediateness. It implies the speaker should already be doing whatever is now overdue, whereas the simple infinitive implies speaker should start doing whatever is now overdue to be done.
As ever, my advice for learners is to stick with the simplest verb form that works. Pure speculation on my part, but I think the Family Guy scriptwriters spend a lot of time deciding exactly what phrasing to use. And while they're definitely not in the business of teaching English to non-Anglophones, they very often deliberately choose "less common" phrasings - sometimes for no other reason than to make the audience pay a little more attention when they don't hear exactly what they might expect. Which happens at a subconscious level.