In what month is the group making its pilgrimage in "The Canterbury Tales"?

You need the first few lines of the whole Canterbury Tales for this - this is an excerpted chunk from the General Prologue:

Whan that Aprill with his shoures soote
The droghte of March hath perced to the roote,
[...]
Thanne longen folk to goon on pilgrimages,
And palmeres for to seken straunge strondes,
To ferne halwes, kowthe in sondry londes;
And specially from every shires ende
Of Engelond to Caunterbury they wende...
[...]
Bifil that in that seson on a day,
In southwerk at the tabard as I lay...

So the answer is right there: it's April. Or at least, it looks like being April, it was only "in that season" that Chaucer stayed in Southwark at the Tabard Inn where he met the pilgrims.

April is the month when things start to grow: the rainy weather of March has brought the shoots out of the ground. And that's the time when people want to go on pilgrimages - and go off to Canterbury.

Hope it helps!