No reason why not, as long as the 14650 is a regular Li-ion or a protected Li-ion, or an IMR or hybrid. Sounds like an unusual cell, the more common size is 14500.
The TR1 will charge any 3.7 volt Li-ion type rechargeable battery (3.7v and 3.6v cells are the same). It will charge to about 4.21 volts fully-charged. It won't reboot a cell that has been discharged too low, so take your battery out of the APV before it gets too low. At around 3.4v is a good choice as this prolongs battery life, though it may be possible to go down to 3v or lower.
Don't use it for a 3.3v Li-FePO4 cell though, it will damage them.