I used to own the XP-60. I haven't used one in a while, but I'll try to go from memory here--so bear with me
Under your "Realtime-Rec Stand-by" screen, all the info such as "measure", "tempo" ,"loop", ect should be here. Quantz. should be right under "Mode". Here you can turn Quant. on or off, and make settings. Make sure it's set to "on". And set to "Grid", then set the "Grid Resolution" to 100%. This should apply quant. to your recording.
Hope this helps,
Squeak
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GEAR: Yamaha MOXF-6, Casio MZX-500, Roland Juno-Di, M-Audio Venom, Roland RS-70, Yamaha PSR S700, M-Audio Axiom Pro-61 (Midi Controller). SOFTWARE: Mixcraft-7, PowerTracks Pro Audio 2013, Beat Thang Virtual, Dimension Le.