Yes, but it's not what you think.
The Dwarf-runes in the Hobbit actually don't come from any of Tolkien's pre-existing works, but are instead Anglo-Saxon runes. This is confirmed by Letter 15:
In any case – except for the runes (Anglo-Saxon) and the dwarf-names (Icelandic), neither used with antiquarian accuracy, and both regretfully substituted to avoid abstruseness for the genuine alphabets and names of the mythology into which Mr Baggins intrudes – I am afraid my professional knowledge is not directly used.
The langauge behind them is therefore Old English.
Subsequent development of the concept (which may be read in Lord of the Rings Appendix E) traces the runes through the following steps:
- The original Cirth of Daeron which were devised for representing Sindarin.
- The Angerthas which were an extension of the Cirth, with most of the work attributed to the Noldor of Eregion in the Second Age "since they were used for the representation of sounds not found in Sindarin"; at this stage we can deduce that the language used was Quenya.
- These were then taken up by the Dwarves of Moria - presumably on account of the friendship between Eregion and Moria - and used for Khuzdul and Westron.
- And finally they were refined further by the Dwarves of Erebor.
This final refinement is, however, not the runes used in the Hobbit, which remain the original Anglo-Saxon runes for the reason given in the Letters quote above. The runes used in the Hobbit are a substitution for what their real forms would have been, and have no basis in the constructed languages of Middle-earth.
The histories given in the Silmarillion note a separate development of the original Cirth carried out by the Dwarves of Nogrod and Belegost in the First Age, but this development presumably stopped when those mansions were destroyed in the War of Wrath.