No; the pattern of final <z> in the singular vs. <ces> in the plural is a modern spelling convention that's unrelated to the Old Spanish distinction between a voiced and voiceless sibilant phoneme (thought to have originally been affricates [t͡s] and [d͡z]). In Old Spanish, both the singular and plural forms of this word were usually spelled with <z>: vezes/uezes (although ueçes is apparently found once) (The Old Spanish Sibilants, by Jeremiah Denis Matthias Ford, 1900, page 10).
Old Spanish spelling used <z> in different places from modern Spanish
According to J. D. M. Ford (Old Spanish Readings, 1911), Old Spanish did not distinguish the voiced and voiceless sounds at the end of a syllable, both being written in this position as <z> (pages xxxvi-xxxvii). In fact, Ford argues that this represented a phonetic neutralization where in word-final position, only the voiceless version of the sound existed (page xxxvi). This would mean that the situation in Old Spanish was actually the opposite of what the modern spelling suggests: the singular vez ended in a voiceless consonant that corresponded to a voiced intervocalic consonant in the plural vezes. So the pronunciations at that time would be something like [ˈβet͡s] vs. [ˈβed͡zes].
The plural vezes had [dz] in Old Spanish because the voiced sibilant was the regular phonetic outcome in intervocalic position of Latin singleton c followed by a /i/ or /e/ that retained its syllabic value*. Thus, we also have Old Spanish fazer from Latin facere, and uezino from Latin vicīnum (first syllable shortened from Classical Latin vīcīnum).
*When Latin /i/ or /e/ was unstressed and followed by another vowel, it was turned into a non-syllabic glide /j/, which merged early on with the preceding consonant and could produce a different outcome from the later palatalization before syllabic /i/ and /e/. In Spanish, Latin intervocalic c + unstressed i/e + vowel sometimes resulted in voiceless ç/c, sometimes in voiced z; it seems a bit difficult to determine which is regular.
Modern Spanish changed spelling to use <ce> and <ci> in most words
After the phonetic merger of the voiced-voiceless sibilant pairs, <ze> and <zi> came to be pronounced the same way as <ce> and <ci>, leading to variation between the spellings.
When the Royal Spanish Academy, founded in 1713, published its first dictionary, the Diccionario de Autoridades (1726-1739), it used a spelling system that did not indicate the historical voicing distinction between [t͡s] and [d͡z]: <ç> was eliminated in favor of <z>, and words that were spelled with <ze> or <zi> in Old Spanish were respelled with <ce> and <ci> (there are only a few exceptional words currently spelled with <ze> and <zi>, such as zeugma).** So Spanish now has the spellings hacer, vecino, veces which don't match the Old Spanish pronunciations.
I think it's possible that the choice to generalize the spellings <ce> and <ci> rather than <ze> and <zi> was influenced by the Latin etymons of the affected words: most of them came from Latin words spelled with <c>. For comparison, Old Spanish had a contrast between intervocalic <b> [b] (as in lobo [lobo]), derived from singleton Latin /p/, and intervocalic <u>~<v> [β], derived from Latin /b/ and /w/; after this contrast was lost, a number of words spelled with <v>/<u> in Old Spanish that had /b/ in Latin came to be respelled with <b>, such as Old Spanish auer [aˈβer] from Latin habeo becoming respelled in modern Spanish as haber. The modern use of <b> in these words seems to be motivated by their Latin etymology.
** Interestingly, the official elimination of <ss> in favor of <s> didn't happen until later, in 1763, and the replacement of <x> with <j> only happened in 1815. ("An overview of the sibilant merger and its development in Spanish", by Eva Núñez-Méndez, Chapter 1 in Sociolinguistic Approaches to Sibilant Variation in Spanish, 2021; page 57)