Early Church Fathers and "Q" Starting with the early church, a large portion of Christian tradition has held the Augustinian hypothesis which believes that Matthew's Gospel was the first to be written, then Mark's, then Luke's, and subsequently John's, with each writer having some knowledge of the others' writing. 

Since the 19th century however, many biblical scholars have rejected Augustinian hypothesis (namely that Matthews Gospel was the first to be written) in favor of Markan priority. On top of this, they assert that Matthew and Luke's Gospels have used Mark's as a source from which both draw virtually identical narrations (see The Synoptic Gospels). Under this hypothesis, those portions of Matthew and Luke that are in agreement but do not appear in Mark are believed to have been borrowed from an alternate source commonly referred to as "Q" 

"Q" as a tangible fragment or codex has never been discovered within the field of archaeology and paleography however. Many biblical scholars defend "Q's" lack of tangible existence due to many portions of scripture being lost or edited. 

However, one group certainly must have recorded an alleged Gospel as important as "Q" would have been, and that group is the Early Church Fathers.

So, do any Early Church Fathers record of another Gospel similar in content to those portions of Matthew and Luke that do not appear in Mark?
As has been discussed in the comments of the OP, the likelihood that sources or "accounts" (Luke 1:1) which the Gospel writers may have used as references, being documented, is scanty at best. That being said, there exists at least some evidence from one church Father that has been used to promulgate the potential of what could be "Q."

The church father in question is none other than Papias of Hierapolis (roughly 60-130 CE). 
Although some sources claim Papias' writings survived through the middle ages, they certainly have not made it into the 20th and 21st centuries.1 
Thankfully, we have some portions of Papias' writings seemingly quoted verbatim by Eusebius of Caesarea in his "Ecclesiastical History".  

Eusebius quoting Papias' writings pertaining to the origins of the Gospel of Matthew: 


  "Therefore Matthew put the logia in an ordered arrangement in the
   Hebrew language, but each person interpreted them as best he could."
  - Eusebius of Caesarea, "Ecclesiastical History"  3.39.14-17


The translation of the word "logia" in the english ("λόγια" in the Greek), is the main point of contention. In non-Christian contexts, the word was traditionally interpreted as "oracles", however 19th century biblical scholarship argued "logia" was better interpreted "sayings." 2 
This interpretation has given credence to the notion that Papias is really referring to a "sayings" gospel like that of the the Gospel of Thomas or "Q" that Matthew incorporated into his Gospel. 

Although not the strongest of arguments, this case does at least give potential to an ECF's acknowledgement of a source used by Matthew that would've contained content similar to what "Q" is theorized to contain. 

Sources:


Harnack, Adolf (1893). Geschichte der Altchristlichen Litteratur bis Eusebius. 1. p. 69
Lührmann, Dieter (1995). "Q: Sayings of Jesus or Logia?". In Piper, Ronald Allen. The Gospel Behind the Gospels: Current Studies on Q. pp. 97–116.