Sheol was the place where spirits dwelled after they died, so the answer is yes in that sense. Among its characteristics, which are variously described:
The dead descend or are made to go down into it; the revived ascend or
are brought and lifted up from it... Here
the dead meet without
distinction of rank or condition—the rich and the poor, the pious and
the wicked, the old and the young, the master and the slave... The
dead continue after a fashion their earthly life. Jacob would mourn
there; David abides there in peace; the warriors have their weapons
with them, yet they are mere shadows. The dead
merely exist without knowledge or feeling.
Silence reigns supreme; and oblivion is the lot of them that enter
therein. Hence it is known
also as "Dumah," the abode of silence; and there God is not praised. Still, on certain extraordinary occasions the dwellers in Sheol
are credited with the gift of making known their feelings of rejoicing
at the downfall of the enemy. Sleep is their usual
lot. Sheol is a horrible,
dreary, dark, disorderly land; yet it is the appointed
house for all the living. Return from Sheol is not
expected. (See article Sheol for biblical refs.)
Thus Sheol was a realm in which the dead, for the most part, lived in a semi-conscious state or worse. But it is hard to know how well acquainted Hezekiah was with the above descriptions. What we do know is that Hezekiah says "Sheol cannot praise You; Those who go down to the pit cannot hope for Your faithfulness." See the context for clarification. NIV does a good job with this IMO even though changes "sheol" to "the grave":
A writing of Hezekiah king of Judah after his illness and recovery..
The grave cannot praise you,
death cannot sing your praise;
those who go down to the pit
cannot hope for your faithfulness.
The living, the living—they praise you,
as I am doing today.
Hezekiah is thanking God for his deliverance from a deathly illness. His words about Sheol reflect the attitude of other Judean poets as well. Psalm 6, attributed to David, is an example:
Have mercy on me, Lord, for I am faint;
heal me, Lord, for my bones are in agony.
My soul is in deep anguish.
How long, Lord, how long?
Turn, Lord, and deliver me;
save me because of your unfailing love.
Among the dead no one proclaims your name.
Who praises you from the grave [sheol]?
So yes, Hezekiah believed in an afterlife, but not one in which people could relate to God and probably not even hope for deliverance. His writing here reflects an attitude that was not uncommon--that God is better served by the living than by those who dwell in Sheol.