I have a pre-beet seed myself. I had the bonus chest on, and when I went to check it at spawn after they were added, I had two beetroot seeds in it. So if you have bonus chests enabled, you might be in luck.
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So if I have explored the surface, I may have a chance underground if I can find a mineshafts?
It will depend on when you did your exploring. Before the first "1.9 update", chests were populated with loot once the chunk they were located in was loaded. Now, chests don't get loot until they are accessed (you can test this by breaking a loot chest prior to opening it--you'll get nothing but the chest).
So, if you explored your world after the first "1.9 update" you've got a shot at beetroot seeds. Otherwise...good luck buddy.