Yes, Psalm 91 starts off with the psalmist speaking and then God speaks at the end.
God selected certain men to write down His words in the Bible. The psalmist is one of those men. And he wrote being inspired by the Holy Spirit.
"All Scripture is inspired by God and profitable for teaching, for reproof, for correction, for training in righteousness..." (2 Timothy 3:16)
How this works in the Bible is that sometimes God gave the writers points to write down and they wrote in their own words (using their own voice). Other times, God dictated verbatim what He said.
You see both of these types of writing in Psalm 91.
First, the psalmist is inspired to tell us that those who trust in God are covered and protected by Him.
"He who dwells in the shelter of the Most High
Will abide in the shadow of the Almighty..." (vs. 1)
Then the psalmist says to God that he himself is one who trusts in Him:
"My refuge and my fortress,
My God, in whom I trust!" (vs. 2)
Next the psalmist is inspired by God to write what happens to others who trust in Him:
"You will not be afraid of the terror by night,
Or of the arrow that flies by day;
Of the pestilence that stalks in darkness,
Or of the destruction that lays waste at noon.
A thousand may fall at your side
And ten thousand at your right hand,
But it shall not approach you..." (vs. 5-7)
And the psalmist prophesies what happens to Jesus, because God caused angels to guard Him on earth:
"For He will give His angels charge concerning you,
To guard you in all your ways.
They will bear you up in their hands,
That you do not strike your foot against a stone." (vs. 11-12)
This verse was actually used against Jesus by Satan later to tempt Him to test God (Matthew 4:5-6).
Nevertheless, we can also apply this to ourselves because no matter what happens to us on earth, if we follow God and don't give up, God will guard our ways and show us His salvation in the end.
To finish up the psalm, the psalmist wrote down God's words verbatim (switching to God's voice).
"Because he has loved Me, therefore I will deliver him;
I will set him securely on high, because he has known My name.
He will call upon Me, and I will answer him;
I will be with him in trouble;
I will rescue him and honor him.
With a long life I will satisfy him
And let him see My salvation.” (vs. 14-16)
So Psalm 91 ends with God promising in His own words the benefits of trusting in Him.
For those who love God, He will give eternal life and let us see His salvation.