There are a lot of good modern string quartets. I have Carter's first two by the Composers Quartet on a single CD. These are in the Nonesuch box. I find them a bit baffling and hard to follow. I really like his wind quintet, but I don't know of any recordings. There's a great performance on YouTube by the Armani Quintet. I also like his piece Steep Steps for bass clarinet. You can find it on the Bridge label. They have a fine series of Carter recordings. But for string quartets, I like modern sounds and dissonance, but I want some lyricism as well. Quartets by Carter, Ferneyhough, and other ultra modernists can be tough going. Lutoslawski's is also pretty strange.
Here's a short list of some 20th century quartets I enjoy.
Hindemith Quartets by Kocian Quartet, and Amar Quartet
Rihm Nos. 1-4 by Minguet Quartet on Col Legno
Dutilleux's quartet on Erato
Zemlinsky Nos. 3 & 4 by Artis Quartett
Wellesz No. 3, 4, 6 by Artis Quartett
Webern- String Works by Artis and Schoenberg Quartets
Norgard nos. 7-10 by Kroger Quartet
Penderecki's quartet on the Vox label.
Bacewicz Quartets on Naxos these are superb, recorded in great sound by the Lutoslawski Quartet.
Karl Hartmann Nos. 1-2 on Nimbus label
Berg Lyric Suite by Emerson Quartet