What Amplifier would you recommend to power KEF Q350’s?

Why in the world would you be willing to spend $1500 to power $550 bookshelves? Your amps should not cost more than your speakers.  
  
I wouldn’t get better than a Yamaha S501. Even if you’d be upgrading speakers in the future, a $550 integrated is easily suitable for even $2000 speakers.

My roommate uses a yamaha A-S301 with his Q300's and its plenty of power for his setup on his desk. Now that's nearfield but the system plays really loud. I would think the A-S501 would be plenty of juice.

I guess it depends what you're looking for in an amp. Do you have a digital or analog front-end? Do you need digital inputs in an integrated or will you get a separate DAC and a purely analog integrated?

I can think of some reasons for spending more on amplification than your speakers. If your amp is over-capable, it allows for a more seamless upgrade to a better speaker in the future. I suspect the Kef's would benefit from a nice amplifier, my LS50's certainty do.  

I believe either the Yamaha A-s501or the Marantz PM6005 would make for a upgrade over the Sherwood and complement your current speakers well. The Yahama has both phono and digital inputs (no USB). The Marantz has phono input, don't recall about digital. Both are well built. 

Yaqin MS-34C....