I have the 2015 Harper Collins paperback edition (ISBN 978-0-261-10348-1); the page numbers in this post are from that edition.
I can't find a specific mention of Durin II, but Part Two Chapter X of The Peoples of Middle-earth (beginning on page 295) is called Of Dwarves and Men. There is a section (beginning on page 301) called Relations of the Longbeard Dwarves and Men. It begins by identifying Longbeard Dwarves as Durin's folk. On page 302, it describes the Longbeards began to deal with Men in Eriador.
There dealings between Men and the Longbeards must soon have begun. For the Longbeards, though the proudest of the seven kindreds, were also the wisest and the most farseeing. Men held them in awe and were eager to learn from them; and the Longbeards were very willing to use Men for their own purposes. Thus there grew up in those regions the economy, later characteristic of the dealings of Dwarves and Men (including Hobbits); Men became the chief providers of food, as herdsmen, shepherds, and land tillers, which the Dwarves exchanged for work as builders, roadmakers, miners and the makers of things of craft from useful tools to weapons and arms and many other things of great cost and skill.
The History of Middle-earth Volume XII, Part Two, Chapter X: Of Dwarves and Men
Page 302 (Harper Collins 2015 paperback edition)
It goes on to say
This system developed slowly, and it was long before the Longbeards felt any need to learn the language of their neighbours, still less to adopt names by which they could be known individually to 'outsiders'. This process began not in barter and trade, but in war; for the Longbeards had spread southward down the Vales on Anduin and had made their chief 'mansion' and stronghold in Moria;
Although Durin II is not mentioned, this is early in the Second Age, so it is quite possible that Durin II was alive at this time (Durin I died before the end of the First Age).