There are three ways to obtain these valuable-looking pieces of lumber. Mayor Thomas will come by your farm on the 2nd of Winter and ask you for a piece of White Grass or some other hard to get item. If you get the item he requests by 7PM on the day he asked you, he will reward you with a piece of Golden Lumber.
Alternatively, Gotz will sell these for 100,000 G each after you have collected 999 pieces of regular lumber in your shed.
The last one is having a 50-59 streak on the goddess' new year game.
The Golden Lumber, while useful on your farm has an incredibly negative effect. It makes for a wonderful fence on your field, as it does not wither or break. You will never have to replace the lumber.
If you put it on your field, a stream of townspeople will come into your farm and scold at you for being rude and inconsiderate the following morning. They will continue to harass you until you take the Golden Lumber off of your property. If you're worried about your friendships with the townsfolk, do not use Golden Lumber on your property. The other townspeople are just jealous at you because you have Golden Lumber on your property.