bubblyrat
You're probably thinking of the Lady Juliana which was the first convict ship with women prisoners - see the book "Floating Brothel" published several years ago.
The first fleet left England in May 1787 and dropped anchor in Botany Bay on the 26th January 1788. The Lady Juliana sailed from Plymouth in July 1789 and disembarked her prisoners on the 11th of June 1790 almost a year later after calling for fresh provisions at Tenerife; Rio de Janiero and the Cape of Good Hope - just as the first fleet had done before her.
I think that the fastest trip out was by a quite small ship called the Amazon which took about four and a half months.