Here goes... (not sure about the maths)
I don't have anything on the Enterprise 1701 Warp engines, or the Enterprise NX-01, but according to the 'TNG: Tech Manual', on the Enterprise-D, Warp 4.5 would be equivalent to a speed in between 102 times c and 214 times c. (between 30,000,000 km/ sec and 64,000,000 km/ sec) But, as I said, Roddenberry recalibrated the Warp scale between the first and second series, so we really need something like the book 'Mr. Scott's Guide to the Enterprise' to work it out.
I think it was designed so that the Enterprise 1701 could visit a new planet each week, with a weeks journey in between to get there. i.e. a weeks journey between inhabited star systems.
In 'Broken Bow' Archer actually says that they will reach Q'noS in 80 hours "will be to Q'noS in about 80 hours... 80 hours Doctor!..." (I can't remember 4 days being mentioned.)
"Neptune and back in six minutes at Warp 4.5" says Trip. That all depends on where the Earth and Neptune are (both on the same side of the Sun or not) but would between 4,350,000,000 km and 4,550,000,000 km. That means that at Warp 4.5 they can travel 12,222,222 km/ sec. That's not very fast at all.
But, they weren't even travelling at 4.5, only at 4.3 and reached 4.4 into the second act of 'Broken Bow'. Archer says that they are travelling at 30,000,000 km/ sec at that point. There is some mistake there straight away, but if we ignore the 'Neptune and back' figure, and ignore all the big confusing numbers, what Archer says exactly matches what I worked out from the 'Tech Manual'. So, I think the idea must be that they are travelling at about 102 times c.
If we assume that, it works out at about 0.28 light years/ day. In 2 days they could travel 0.55 light years. In 80 hours about 0.93 light years could be covered. The nearest star (Proxima Centuri) is 4.3 light years. Barnard's Star is 6.0 light years, Wolf 359 is 8.1. You have to travel over 12 light years before you reach more than 25 stars. So, in 80 hours they would barely make it out of our solar system.
I assume (from 'Best of Both Worlds' TNG) that the Klingon Empire is further away that Wolf 359, so it is at least 8.1 light years away, most likely Q'noS orbits one of the stars about 11 light years away, so it would take them at least 1 month to reach it.