"strong consciousness" is correct and usable in written English.
You can use it to refer to a strong sense of awareness or alertness. For example, "The leader acted with a strong consciousness of world affairs."
The other was a sense of patriotism, a strong consciousness of our American identity.
Thus, although Japan early adopted a version of the German Civil Code, it did not adopt the Germans' strong consciousness of legal rights.
But Swedish women, backed by a strong consciousness of women's rights and a history of a very public discussion of the scourge of sexual violence, may be more willing than most to look to the law for help.
Most of them were aware that protection along a route of more than fifty miles of hostile country could not be absolute (on the night before the march, a student who had come here from Boston University was slashed across the cheek with a razor blade), yet few of the thirty-two hundred marchers who set out on Sunday morning seemed to have a strong consciousness of risk.
The urban population has a strong consciousness.
When people saw the portraits they said, "lovely children, but who are they?" The photographs showed lovely children, with a strong consciousness: this is who I truly am.
On the other hand, it may strengthen the union, leading to a stronger consciousness of the importance of marriage and increased efforts to ensure marriage stability.
All of the organised world religions now have a strong environmental consciousness.
She has a strong social consciousness that she was able to translate into her art".
Some celebrities don't have the strong social consciousness you have.
The strong group consciousness of dockworkers in African countries made them among the first to take collective class action.
