Yes, the 45th Missouri was part of the garrison at Jefferson City on October 7, 1864. Here is Major Lewis Boutell's report of their action (he was commanding the regiment).
Number 26. Report of Major Lewis H. Boutell, Forty-fifth Missouri Infantry.
HDQRS. FORTY-FIFTH MISSOURI VOLUNTEER INFANTRY,
Jefferson City, October 8, 1864.
CAPTAIN: I have the honor to report that Companies F and I, under command of Adjutant Hill, were, by order of General Brown, sent forward on the road leading from the fair-ground to support a section of artillery which was to be sent there. On reaching the road beyond the fair-ground Adjutant Hill was ordered by a general officer, name unknown (but think General Sanborn), to deploy one of the companies as skirmishers and to relieve his line, which he did. The road was filled by cavalry of the Sixth Missouri State Militia. After deploying one company as skirmishers he was ordered by a colonel of cavalry to send the company retained as reserve on the flank to prevent a flank movement. This was done, but it was too late, as the enemy were already on the flank. The cavalry left in disorder, and finding the infantry alone and unsupported, the order was given to fall back. The men, considering they had to be formed under fire and were new soldiers, behaved well. Loss as far as known was 3 killed, 9 wounded, and 4 missing in action.
I am, very respectfully, your obedient servant,
L. H. BOUTELL,
Captain JAMES H. STEGER,
Assistant Adjutant-General, District of Central Missouri