No. Plenty of beings are conscious but are not moral. Your cat Bessie has no idea what the heck morality means and will happily torture mice and birds for the fun of it. For lots of humans morality is a very abstract concept and they don’t really understand it or work with it in their ordinary lives.
Morality comes into play once you can realize that there’s a bigger picture than just your own life. For many people this concept is religiously-defined, but for almost as many it’s not. Let’s take an example from film:
In case you haven’t seen Pulp Fiction, Jules Winnfield, played brilliantly by Samuel L. Jackson, has a crisis of conscience in the film in which he discovers both God and morality, in one fell swoop. Shortly after this frame, he survives a point-blank shooting in which an entire magazine was emptied at himself and his partner, Vincent. Unlike the three people he and partner killed, Jules and Vincent emerge completely unscathed.
Jules, to hear him tell the story later on, had just gone about his gangster life without a care in the world for how his actions affected others. He even claimed his violence in the name of crime was moral when he recited what he thought was a Bible verse shortly before his mob executions. As if God Himself sanctioned Jules’ murders.
This came to a stop when God appeared in Jules’ life and made His will known. Jules decided he was done with gangster work and that he would “walk the Earth.”
Naturally, people miss at point-blank range all the time, particularly those untrained with high-caliber weapons, as Vincent pointed out at the time. It doesn’t matter whether Jules’ beliefs were justified, what matters is that Jules very clearly did not have a moral sense before the event, and had, suddenly, to develop one, after it became abundantly clear to him that he needed one. It should also be noted that Vincent did not survive the events of the movie owing to his stupidity, and Jules did.
We see the initial results of his moral journey when he, rather than killing Ringo and his girlfriend for robbing the diner he and Vincent were having lunch in, gives him the money in his wallet and lets him leave with the rest of the diner’s customers’ money. “Normally, both of your sorry asses would be deader than ****ing fried chicken by now, but you happened to pull this shit while I'm in a transitional period, so I don't wanna kill you. I wanna help you.” You really should go watch it if you haven’t.
Jules’ journey can be contrasted with another character who has moral moments in the film. Butch, portrayed serviceably yet iconically by Bruce Willis, has many moments in the film where he makes personally-dangerous decisions to put himself in harm’s way for honor, his first exposure to which was given to him when Captain Koons tells him in excruciating detail what his father had to go through so that Bruce could wear his gold watch.
Butch might never have developed a moral compass had Koons never kept his word to Butch’s father and presented him with the watch. But we can see throughout the film that Bruce simply lives in a different moral world than Jules could ever dream of. Bruce doesn’t punish Fabienne for forgetting to pack his father’s watch and instead blames himself, and, at great personal risk, goes back to his apartment to retrieve it.
Even though the gangster he betrayed tried his hardest to kill him, Bruce didn’t leave Marcellus to his (well-deserved) fate at the hands of Zed, he strode back into the lair of the beast and saved him. In return Bruce gets to live his life having betrayed the trust of a notorious gangster without having to look over his shoulder every step.
This conclusion is a bit outside of the scope of the question, but it’s useful to show how morality develops. Morality germinated in Jules at the point in his life the film portrays and you get to see how that works. But Butch has been proceeding along his moral journey since he was a kid, and we get to see what that has gifted him with.
Morality allows you to express yourself and your unique essence without hurting others in the process. How much of it you have is how much less your actions hurt. Other people notice and recognize your morality and learn from you. They hurt others less by their future actions. And then they save you from consequences you would otherwise have had to deal with by virtue of their lesser sense. God showed mercy to Jules the same way you show mercy to your precious Bessie when she murders half your neighborhood’s birds.