Depends on the rune (but yes)
The rules for Specific Magic Weapons state that:
These weapons have abilities far different from what can be gained by simply etching runes. A specific magic weapon lists its fundamental runes, which you can upgrade, add, or transfer as normal. You can't etch or transfer any property runes onto a specific weapon that it doesn't already have, and you can't remove its property runes.
Note the "doesn't already have" part, which implies by omission that you can etch or transfer a rune that the specific weapon already has.
To understand this we need to look at rules for Reading Items, specifically those about Multiple Types and Crafting Requirements:
If multiple types of an item exist, the title line gives the minimum level followed by a plus symbol (“+”). The description includes information on the base version of the item, and the Type entries at the bottom of the stat block lists the specifics for each version, including the level, Price, and any modified or added abilities of the different types. For some items, the types listed are upgrades to the base item. For other items, such as aeon stones and wondrous figurines, each type is distinct from the others.
The GM might allow you to Craft a permanent item from a lower-level version of the same item as an upgrade. For example, you might upgrade a bag of holding from a type I to a type II bag, but you couldn't upgrade a clear spindle aeon stone into an orange prism aeon stone. The cost for this upgrade is the full difference in Price between the items, and the Crafting check uses a DC for the item's new level.
Currently, there are no property runes with types that are distinct1, but if such a rune were to exist on a specific weapon, you would not be able to upgrade it.
On the other hand, there are property runes with types that are upgrades, and those you can, well, upgrade.
For example, Shock has two types: a base version, and a greater version that is clearly an upgrade of the former. You could etch either type of Shock onto a specific weapon like Storm Flash, because it already has a Shock rune. If you etch the greater version, you can upgrade the existing Shock, as described above and in the rules for Runes:
An item can have only one fundamental rune of each type, though etching a stronger rune can upgrade an existing rune to the more powerful version (as described in each rune’s entry).
Although that is specifically about fundamental runes, it applies to property runes too, as a later paragraph states that:
You can upgrade a property rune to a higher-level type of that rune in the same way you would upgrade a fundamental rune.
- Even a rune like Bane, which only works on one creature type, still doesn't have the entries for multiple types.