Embouchure changes will do it. Start with practicing on your mouthpiece alone. When you learn to play an octave on your mouthpiece without biting down on the reed you will have the basic idea down. Then you just have to practice on the full instrument.
Playing quarter tones is easy. My beginning bands did it all the time. I agree with the above advice, but to answer the original question this is what I found on the web.
Quarter tone fingerings 1st octave
As always excellent professional advice from jbt.
For country sax listen to Bill Evans (sax) with Bèla Fleck
For Bill Evans would be the Soulgrass album
and for Bèla Fleck and the Flecktones just look for anything where Bill Evans is on there.
May not be your idea of "Country" but "Yee haw and Hallelujah".
The bending of notes is easiest higher up, so higher saxophones like soprano can work really well for arabic and/or klezmer styles. As mentioned, the lip/throat techniques are probably more useful than actual 1/4 tone fingerings.
microtonal fingerings for woodwind have been around for a few decades, Bruno Bartolozzi's New Sounds For Woodwind may have been the first publication that covered this area, but alas the saxophone isn't included in that book and most quartertone compositions for woodwind seem to be for clarinet or flute.
Joe Maneri was the pioneer of microtonal jazz sax playing, but there aren't any good examples of his playing freely available - most of the stuff on youtube is of his clarinet playing.
So here's what can be gleaned from the internet