Motherboard wise you are completely fine.
Assuming CPU is an i5 or better and you have 8gb of ram or better you are fine in that department.
Power Supply will for absolutely certain not support a 1070. You will need at bare minimum 450w (500w prefered) quality power supply. I cant stress "quality" enough, going cheap and buying a $30 PSU will leave you with a fried GPU, motherboard and possibly more. Seasonic and XFX are good choices all around; EVGA, Rosewill, Corsair has some good and some blah units.
The other problem you may run into is airflow. OEM (Acer, Dell, HP, etc) office Desktops are not designed with gaming in mind and thus they have top mounted PSUs and not enough fans to provide the kind of cooling high power gaming...