Can someone married to a EU citizen work for a month from another country?
Yes, if your EU citizen spouse accompanies you during that period then you, as a family member, may exercise your EU citizen spouse rights from the freedom of movement directive.
Tax will be based on national law and tax agreements between the different countries.
If you live and are paid in Germany, you still pay german tax for periods up to 6 months when working abroad.
Italian citizens are liable for taxation for their world wide income until they are registered in the A.I.R.E system. Registration is only required after 6 months abroad.
The German page below states explicidly that employment (for both) is also true for the first three months. This then should be true for all EU countries.
Information for EU citizens - Federal Ministry for Economic Affairs and Climate Action
Entry and residence
Are you a citizen of an EU member state planning to seek employment, study or complete vocational training in Germany? Thanks to the freedom of movement, you can travel to Germany without a visa or a residence permit and stay here for a period of three months. The only documents you will need for entering the country are either a valid passport or an ID card. There are no restrictions regarding access to employment and self-employment in Germany for you or your family.
The same applies if you are a citizen of Switzerland or the European Economic Area – i.e. Liechtenstein, Norway or Iceland.