I checked the Washington State Digital Archives but didn't find birth information for William. I didn't expect to find it since the date was so early.
You'll be lucky to find any record of his birth. You'll have to contact the County Recorder where you think he was born. Adams County Recorder is at wei.secstate.wa.gov/adams/Pages/abouttheauditorsoffice.aspx. Ask about a charge for them to look it up.
Are you sure he was born in Washington Territory? I think I found the family in Camas Creek, Nez Perce, Idaho in 1880. thomas Heyden [corrected to Hayden] b. MO, 28, farmer. Ann, wife, b. OR, 22. William A., b. OR, 2. Bertha, b. ID, 8/12ths.
A birth in Adams Co. in Washington in 1875 would be very unlikely, however, a white birth in Oregon that early wouldn't be unlikely at all. Successful Searching!