Is a 1x6 enough to secure a wall mount sink?
I would use something wider than a 1x6 - why not?  Even if the mounting plate is smaller, the wider board will be easier to hit, and will be stronger.  

If yellow pine or douglas fir are available they are a lot stronger than white pine.  

I don't trust toe-nail attachments with skinny boards.  The old school way to do it would to be notch the studs so the board lays in the notch, then face nail (or better, face screw) it into the studs. 
 But that's a bit of work, not too bad with an oscillating tool but still.  

The easier alternative would be to use an angle bracket.  For example, the Simpson Strong Tie L70Z is not made for this exact purpose, but I think it would be suitable, and it would be easy to make a very strong attachment.