I'm listening to the radio version of The Hitchhiker's Guide to the Galaxy and earlier it was mentioned that a girl on Earth discovered something that will make everything alright, and before she could tell anyone over the telephone about it, the Earth blew up.
Later, Deep Thought mentions that Earth's purpose to find an answer was going to get answered in 5 minutes if it wasn't blown by the Vogons.
Is this referring to the girl's discovery?
Yes. This is made explicit in the fourth book, So Long And Thanks For All The Fish.
The girl is Fenchurch, whom Arthur meets when he returns to the restored Earth. She remembers everything building up inside her over a course of days, until it came to a head when she was sitting in a cafe in Rickmansworth. But she no longer remembers what the answer actually was.
It is hard to tell, since the following books make the issue way more twisted.
From the POV of the first book alone I would say yes, that what the girl thought was the answer.
But in the following books you learn that:
1 If the girl was right, given the nature of the question and the answer, if the Vogons had not destroyed Earth probably something else would have caused the answer to not be delivered (although one can argue that, since the girl probably did not know the question, there were options for the universe as we know it surviving some time more).
2 A long time ago, all of the experiment was wrecked by the unexpected arrival of the Golgofrinchans. Of course, there is still the possibility that some Golgofrinchan found the answer by pure luck.
You ask about the original radio series, not the books, and the answer is yes.
The second radio series (Episode 12) it is revealed that Zaphod Beeblebrox for planning Earth's destruction in order to avoid the ultimate question becoming known. At the time of the destruction, the disruptive effect of the Golgafrinchan B-ark was not known.
Incidentally, it is Slartybartfast who refers to the Vogon destruction of Earth as a "shocking cock up", not Deep Thought.
There are five series of the original radio show. So far as the radio show is concerned, the answer to your question is no.
Deep Thought does mention that Earth's computer program, to find the answer to the ultimate question, was terminated 5 minutes too soon (five minutes before the moment of read-out) due to the destruction of Earth by the Vogons. But this has nothing to do with the girl's discovery.
According to radio series one, the girl (later, in series 3, she is identified as Fenchurch, and becomes Arthur's girlfriend) has only realised how the Earth can be made a good and happier place; she has not discovered anything relating to the computer program, nor about the ultimate answer.
... then, one Thursday, nearly two thousand years after one man had been nailed to a tree for saying how great it would be to be nice to people for a change, a girl sitting on her own in a small café in Rickmansworth suddenly realized what it was that had been going wrong all this time, and she finally knew how the world could be made a good and happy place. This time it was right, it would work, and no one would have to get nailed to anything.
In fact, she never tells anyone her solution, not even Arthur. So we never get to know why she got so excited.
And in Fit the Sixth we learn that the computer program got cocked-up by the arrival on Earth, in its prehistoric dawn, of the Golgafrinchams on their B-Ark -- causing the original cavemen to die out, who were a key part of the computer program that was designed to find the answer to the ultimate question (of life, the universe, and everything).
Bit of a waste of time for the Vogons, blowing the planet apart, when the computer program had in fact been wrecked 2 million years earlier.
