No--it's a different one: words by Alexander Maclagan (1811-1879), music by J. C. Grieve, born 1842; here from John Greig's Scots Minstrelsie of c. 1900.

1.
Come let us climb auld Arthur Seat,
When summer flow'rs are blooming;
When golden broom and heather bells
Are a' the air perfuming.
When sweet May gowans deck the braes,
The hours flee past fu' cheerie,
Where bonnie lassies bleach their claes
Beside the Wells o' Wearie!
[chorus:]
The bonnie Wells o' Wearie!
The bonnie Wells o' Wearie!
Come let us spend a summer day
Beside the Wells o' Wearie!

2.
The "Lily o' St. Leonards" there
Oft spent a sweet May morning,
Wi' gowans gay and sweet blue-bells
Her golden locks adorning.
And there the "Laird o' Dumbiedikes"
Aft gaed to woo his dearie,
And watch his fleecy flocks wi' care,
Beside the Wells o' Wearie!
[cho.]

3.
There Scotland's Queen in stormy times
Forgot her saddest story;
There brave Prine Charlie led his clans
To deeds o' martial glory.
When Johnnie Cope, wi' a' his men
Were scatter'd tamplinteerie,
There Scotland's banner proudly waved
Beside the Wells o' Wearie!
[chorus]

4.
Then let us hail auld Arthur Seat:
Like Scotland's rampant lion,
It tow'rs, a wonder o' the world,
The wildest storms defyin'.
Wi' dauntless front 'neath summer skies,
Or wintry blasts sae dreary,
It stands in peace or war to guard
The bonnie Wells o' Wearie!
[chorus]

5.
O lang may bonnie lassies fair
Wi' Nature's charms around them,
Still bleach their claes on flow'ry braes,
Wi' nae sad cares to wound them!
Lang may her sons 'mid fairy scenes,
Wi' hearts richt leal and cheerie,
Still meet to sing their patriot sangs
Beside the Wells o' Wearie!
[chorus]

That chorus is deliciously laid out in four-part harmony, and this song is (was) a great favorite for sentimental harmonising in the pubs of the West of Scotland, let alone the east. The Lily of St Leonard's and the Laird in the 2nd verse are characters in Scott's "Heart of Midlothian" The Wells themselves are (or used to be) at the south of the Queen's Park, Edinburgh; Arthur's Seat is the name of a prominent hill nearby.