Graft Surgeon's second ability doesn't cause you to move counters from Graft Surgeon onto the target creature. Rather, you put the same number of each kind of counter Graft Surgeon had when it died onto the target creature.

Graft Surgeon's second ability puts all counters that were on Graft Surgeon onto the target creature, not just its +1/+1 counters.

If Graft Surgeon has -1/-1 counters on it when it dies, that ability will include those as well. This may result in the recipient of the counters dying.

If enough -1/-1 counters are put on Graft Surgeon at the same time to make its toughness 0 or less, its second ability will see all of the +1/+1 counters it had when it died as well as the -1/-1 counters it had, and an equal number of each of those types of counters (plus any other applicable counters) will be put onto the target creature.

In some unusual cases, you may end up putting the appropriate counters on more than one permanent. For example, if you control The Ozolith (a card from the Ikoria: Lair of Behemoths set) when Graft Surgeon dies, you'll put the appropriate number of each kind of counter onto both The Ozolith and the target creature.