Is there Cross-country skiing available anywhere in Heilongjang Province?

Yes, some skiing resorts there have cross-country trails. Here below is two for reference:
Maoershan Skiing Resort: Maoershan Town, Shangzhi City in Heilongjiang
Changshan Skiing Resort: Yanshou County in Heilongjiang

If you are a professional skier, ask help from local skiers. They usually know some good sites not known by the public yet. Have fun~