First of all, Picos de Europa are a mountain range that is quite long (~20km) and without a specific location it is hard to advice. However, for the sake of this answer, I just assume that you want to go to the Refugio Delgado Ubeda, a kind of nice hut in the center of the mountain range.
The whole path from San Sebastian via Unquera to the refugio is around 319 km. No way that you can manage that by hiking only in 3 days. However, the first part of the route from San Sebastian until Unquera follows more or less the A-8 from San Sebastian to Gijon. There it should be possible to get a ride to shorten your hike a little bit. I tried hitchhiking in this area a couple of years ago, and if you focus on the country roads instead of the highways, it should not be too much of a hassle to get a ride.
Then in Unquera, you get into more rural areas. There will be less traffic, and therefore, it will take longer to find someone to get a ride. However, in my experience, people are way more eager to help a hiker in rural areas than near cities. So it should also be possible to make it.
So in summary, yes, I think it is definitively possible. However, let me add some personal notes: I would be very surprised if you manage to hike more than 40 kilometres a day for three days in a row. So this would mean, even if everything goes very well, you still have to hitchhike for almost 2/3 of the whole route. Why don't you focus on part of the journey and e.g. spend a day in San Sebastian, then take public transport to Unquera, and then hike the rest (~50 kilometres) in two days?