I believe the below, bolded and italicized, are all five uses of "quest." (Formatting/line breaks aren't entirely intact.)
24.
And yet till All Hallows with Arthur he lingered,
who furnished on that festival a feast for the knight
with much royal revelry of the Round Table.
The knights of renown and noble ladies
all for the love of that lord had longing at heart,
but nevertheless the more lightly of laughter they spoke:
many were joyless who jested for his gentle sake.
For after their meal mournfully he reminded his uncle
that his departure was near, and plainly he said:
‘Now liege-lord of my life, for leave I beg you.
You know the quest and the compact; I care not further
to trouble you with tale of it, save a trifling point:
I must set forth to my fate without fail in the morning,
as God will me guide, the Green Man to seek.’
44.
Then was Gawain delighted, and in gladness he laughed:
‘Now I thank you a thousand times for this beyond all!
Now my quest is accomplished, as you crave it, I will
dwell a few days here, and else do what you order.’
The lord then seized him and set him in a seat beside him,
and let the ladies be sent for to delight them the more,
for their sweet pleasure there in peace by themselves.
For love of him that lord was as loud in his mirth
as one near out of his mind who scarce knew what he meant.
57.
Soon these cried for a quest in a covert by a marsh;
the huntsmen hailed that first heeded the scent,
stirring words he spoke to him with a strident voice.
The hounds then that heard it hastened thither swiftly,
and fell fast on the line, some forty at once.
Then such a baying and babel of bloodhounds together
arose that the rock-wall rang all about them.
73.
She unbound a belt swiftly that embracing her sides
was clasped above her kirtle under her comely mantle.
Fashioned it was of green silk, and with gold finished,
though only braided round about, embroidered by hand;
and this she would give to Gawain, and gladly besought him,
of no worth though it were, to be willing to take it.
And he said nay, he would not, he would never receive
either gold or jewelry, ere God the grace sent him
to accomplish the quest on which he had come thither.
100.
Delight there was awakened, when the lords were aware
that good Gawain had returned: glad news they thought it.
The king kissed the knight, and the queen also,
and then in turn many a true knight that attended to greet him.
About his quest they enquire, and he recounts all the marvels,
declares all the hardships and care that he had,
what chanced at the Chapel, what cheer made the knight,
the love of the lady, and the lace at the last.
24.
And yet till All Hallows with Arthur he lingered,
who furnished on that festival a feast for the knight
with much royal revelry of the Round Table.
The knights of renown and noble ladies
all for the love of that lord had longing at heart,
but nevertheless the more lightly of laughter they spoke:
many were joyless who jested for his gentle sake.
For after their meal mournfully he reminded his uncle
that his departure was near, and plainly he said:
‘Now liege-lord of my life, for leave I beg you.
You know the quest and the compact; I care not further
to trouble you with tale of it, save a trifling point:
I must set forth to my fate without fail in the morning,
as God will me guide, the Green Man to seek.’
44.
Then was Gawain delighted, and in gladness he laughed:
‘Now I thank you a thousand times for this beyond all!
Now my quest is accomplished, as you crave it, I will
dwell a few days here, and else do what you order.’
The lord then seized him and set him in a seat beside him,
and let the ladies be sent for to delight them the more,
for their sweet pleasure there in peace by themselves.
For love of him that lord was as loud in his mirth
as one near out of his mind who scarce knew what he meant.
57.
Soon these cried for a quest in a covert by a marsh;
the huntsmen hailed that first heeded the scent,
stirring words he spoke to him with a strident voice.
The hounds then that heard it hastened thither swiftly,
and fell fast on the line, some forty at once.
Then such a baying and babel of bloodhounds together
arose that the rock-wall rang all about them.
73.
She unbound a belt swiftly that embracing her sides
was clasped above her kirtle under her comely mantle.
Fashioned it was of green silk, and with gold finished,
though only braided round about, embroidered by hand;
and this she would give to Gawain, and gladly besought him,
of no worth though it were, to be willing to take it.
And he said nay, he would not, he would never receive
either gold or jewelry, ere God the grace sent him
to accomplish the quest on which he had come thither.
100.
Delight there was awakened, when the lords were aware
that good Gawain had returned: glad news they thought it.
The king kissed the knight, and the queen also,
and then in turn many a true knight that attended to greet him.
About his quest they enquire, and he recounts all the marvels,
declares all the hardships and care that he had,
what chanced at the Chapel, what cheer made the knight,
the love of the lady, and the lace at the last.
