first some introductory information (and no, you don't have to read everything - the intro, and skim some of it would likely be enough)
The fstab entry only tells the operating system what you want done during a boot. Here you are telling it that you want the logical partition (or volume as a virtual disk is thought of) mounted on the /home directory. (I will assume that the missing "/" at the beginning (and the missing "u" in usrquota) are typographic errors, the line should be "/dev/mapper/vg_ayk00-lv_home /home ext4 defaults,usrquota 1 1"). This is assuming the volume has a filesystem on it.
Assuming this has worked, the quotacheck commond to use would be "quotacheck -c /home" or "quotacheck -vcum /home". You would not be using the "g" option as you did not enable group quotas in the fstab file (it would be "usrquota,grpquota" in the fstab line if you want both).