The conversion note field can be updated with the API, however, this is not accomplished with the “updateConversion” field, but with an auxiliary method called conversionMeta.
Here’s an example API call to update a conversion note:
http://api.hasoffers.com/v3/ConversionMeta.json?Method=update&NetworkToken=API_KEY&NetworkId=DEMO&data[conversion_id]=123456&data[note]=NOTE HERE
Note that the ID field being used is the conversion ID, not the transaction ID. You can obtain the conversion ID from another API call.