No, not really. If it is patented, the owner of the patent might charge a lot for it, or they might give out licenses cheaply out of the goodness of their hearts. If it is not patented, it might be cheap if there are many manufacturers, or it might be expensive if there are few manufacturers (for whatever reason). It might also be intrinsically expensive to manufacture, in which case the result will also necessarily be expensive.
So really, the questions are nearly completely unrelated.
Although in this era of modern biotechnology, especially for a 3rd World targeted vaccine, it would be appalling to develop one that was expensive to manufacture. I.e. nothing like the standard old flu vaccine, which is grown in the membranes of chicken eggs, yielding about 3 doses from each egg. Compare to e.g. the Protein Sciences method, which uses genetically modified insect cells in a bioreactor, which as I recall produces about 100,000 doses in a week or so (although it normally takes another 5-7 weeks to finish the vaccine once a batch of raw antigen is done).