It's not that simple. Your quote is from Isaiah 7:14 and its meaning depends on how the Hebrew word 'almah is translated. Its meaning (which could either be taken as "young girl" or "virgin") has been debated for centuries, and a BBC programme doesn't exactly have much scholarly weight in deciding what the translation should be. Another point is that the original translation of the Old Testament into Greek was done by Greek-speaking Jews, not by Greeks. (See here for more information on that subject.)
Also, that passage from Isaiah isn't an account of the birth of Christ, as it was written centuries before. It is believed by many to be a prophecy of his birth, but in fairness that's not the only way of interpreting the passage, nor even necessarily the most obvious way of interpreting it in context.
On the other hand the gospels, or at least the two that mention Jesus' conception and birth (Matthew and Luke), are pretty clear that Mary was a virgin and that her child was "conceived by the Holy Spirit", however the Isaiah passage is translated. Whether or not you choose to believe in the factual accuracy of the gospel accounts is, of course, up to you, but that is the textual evidence and in my opinion it stands even if one accepts an alternative interpretation of the Isaiah passage.
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