I think the answer to this is yes, more or less, if they were a learned monk or a scholar.

Example from Bede's Ecclesiastical History: " The blessed Alban suffered death on the twenty­second day of June, near the city of Verulam". Original Latin text: "Passus est autem beatus Albanus die .XXII. Kalendarum Iuliarum iuxta ciuitatem Uerolamium".

He talks about years like this: "In the year of our Lord 394, Arcadius, the son of Theodosius, the forty­third from Augustus, taking the empire upon him, with his brother Honorius, held it thirteen years". Original Latin: "Anno ab incarnatione Domini CCCXCIIII Arcadius filius Theodosii cum fratre Honorio, XLIII ab Augusto regnum suscipiens, tenuit annos XIII".

Often he might just mention the saint's day on which such and such an event happened, such as the death of an important person.

I'm sure also there were wide sections of society in medieval Europe who would only have been aware of the agricultural calendar and the feast days of the saints.

This is an inscription of the Bulgarian king Ivan Asen:

Original (Middle Bulgarian or Old-Church Slavonic, not sure):

В(ъ) лѣто ϛ ѱ л и Інд(икта) Г азъ Іѡ(аннъ) Асѣнь Въ Х(риста) Б(ог)а вѣрны Цръ и самодръжецъ блъгаромъ с(ы)нъ старого Асѣнѣ црѣ

Modern Bulgarian:

В лето 6738 [1230], индикт 3. Аз Йоан Асен, в Христа Бога верен цар и самодържец на българите

Translated it would mean:

In the year 6738 [1230], indict 3. I Ioan Asen, in Chirst God faithful king and autocrat of the bulgarians...

So as you can see, they could use both the count since the creation of the world (as calculated from studying the bible), and the years since the birth of Christ.

Also note the usage of indiction - https://en.wikipedia.org/wiki/Indiction.

Hey, I know the question has been answered already but I wanted to give and ancient Irish perspective on it. Here is what was written by the Irish monks in the Annals of Ulster for the year 1109.

Kalends of January sixth feria, twenty-sixth of the moon. AD 1109, and Easter fell on the seventh of the Kalends of May 25 April and Low Sunday on the second day of summer 2 May, and the feast of Mo-Cholmóc of Liath on Shrove Saturday 13 March. Gilla Ailbe ua Ciarmaic, king of Áine Cliach, died.

It just shows that they knew well how to calculate the date of the year even this far back.

Taken from this source http://www.ucc.ie/celt/publishd.html

In the Islamic medieval tradition the dates are generally indicated similarly to the way they are today. "On the first of the month Sha'ban in the year 850" corresponds to October 31, 1446.

However, in certain literary contexts, the date is indicated by a chronogram, which is a short line of poetry in which the numeric values of each letter in the Perso-Arabic alphabet (a=1, b=2... gh=1000) are added together to produce a date. These could often be quite artful.