The Nature of Definitions
Can we define the act of defining?
Yes, and in a multiplicity of ways. Definition is essentially an association between a term and other terms, and is related to the intension and extension of terms.
Definitions might be stipulative, lexical, precising, theoretical, or persuasive if classified by function.
You seem to be concerned that lexical definitions are inherently and ultimately circular in a dictionary like the OED. Words in a soft dictionary indeed are defined in terms of one another, and that might create the illusion that there's no real meaning. Yet, meaning can be conveyed in a variety of ways in language alone. For instance, language has phonological, syntactic, and semantic structure at a minimum, and words convey meaning at the morphological level. Semantics is a complicated affair, and the philosophy of language has shown that it's not as simple as words are containers of meaning.
If you have an interest in definitions and meaning, start with an introductory logic book which covers definitions such as Hurley's _A Concise Introduction to Logic... as you get more advanced, dip into linguistics such as Jackendoff's _Foundations of Language. Become familiar with the basics such as the genus-differentia method before looking towards more philosophically complicated versions such as recursive definition and explication.
Defining Doesn't Require a Definition of Defining
one cannot define definition because definition has not been defined yet!
There are some definition methods that don't require definition. Consider the ostensive definition. If you want to teach an ape sign language, you could get it to associate a gesticulation in sign language by pointing. Note, the chimp requires no definition of definition to associate the meaning of the symbol with the referent. This is how a child learns the meanings of word: context within experience.
Object and Metalanguages
This is why I think there is a fallacy somewhere as scientists and philosophers are already defining concepts in their articles. However, this might be due to a common sense of the definition of the act of definition which means there is no objective platform as common sense is subjective. Would you please inform me of any fallacy or develop/justify this argument?
Your error in reasoning confuses the ability to use language with metalanguage. Let's take a phone system, for instance. When you talk to an automated voice system, it can understand 'yes' and 'no' when it asks you questions, and act appropriately. However, what it cannot do is define 'yes' and 'no'. Human beings often show the same signs of a lack of reflection when acquiring language. Ask a young child what is a banana, and they might reply 'A banana is a yellow piece of fruit'. But, then ask them, 'by what way did you define banana, by genus-differentia?' You might not get a response. That is because using an object language and using a metalanguage are two DISTINCT uses of language, with the latter being more sophisticated than the former. For more information, read the article on use-mention distinction.
The Turing Test is an example of an operational definition that defines human-level intelligence by using the ability of a system to convince a human being that it understands language like a human does. These systems inevitably perform poorly because computers, while able to use definitions, seldom are able to respond to definitions ABOUT definitions or show other signs of metalinguistic awareness. By using a series of probing questions, most systems can be shown to lack comprehension of a metalanguage that comes naturally to even children.