We know that organic compounds are compounds which contain carbon atoms. I want to know whether all biological macromolecules (such as proteins, lipids, carbohydrates, and nucleic acids) are organic compounds, or are only a subset of them considered organic?
The definition of "organic" is somewhat vague: it can mean containing carbon, or it can mean coming from living things, particularly as defined by the United States Department of Agriculture.
To further confuse the issue, some carbon compounds, such as sodium carbonate or tungsten carbide, are sometimes defined as inorganic.
So state a particular definition, and examine those compounds according to that definition.
There is a field of chemistry referred to as bioinorganic chemistry.
While all the biological macromolecules are organic, if a metal-ion-containing protein (metalloprotein) is being studied, especially if the focus is upon the metal ion, such as oxidation or reduction of the metal ion, the term "bioinorganic" is often used.
There are a number of different definitions of organic compounds existing, from the oldest (anything that comes from living beings (and is assumed to have been created by a vis vitalis)) to a few contemporary ones. So the definition of organic compound is rather fuzzy, with edge cases such as trifluoroacetic acid sometimes belonging to the organic realm and sometimes not.
However, I am not aware of any definition of organic compound that would exclude the biomacromolecules and deem them ‘inorganic’. So unless your definition of organic is extremely exotic, they are included. (And I cannot imagine an exotic definition of organic compound that would not include lipids, anyway.)