Oh, for sure. As long as "morality" can be defined as adherence to a code other than that defined by the Bible and Christian teachings. Some fundamentalists teach that adherence to God's law is the only morality; "the fear of the Lord is the beginning of wisdom". However, many of the most well-known and well-studied moral theorists of history were Christians who endeavored to explain the morality of the Bible in ways that didn't depend on the divine authority, so to convince nonbelievers that the Law should be followed. Others were those who rejected Christianity, or theology outright, and attempted to solve the problem of the source of morality in the absence of divine authority.
Hedonism, for example, is a moral code stating that the action that causes the most pleasure and the least pain for all involved is the right one. It was originally posited by Democritius, who developed these principles based on the fundamental tenet that there is no afterlife; what we have here is all there is, so we should make it the most pleasurable existence possible for all. Although the term is now more often used to refer to what is more accurately "Moral Nihilism" ("if it feels good, do it"), the concepts of "net pleasure" and "net pain", when applied to the collective, would result in several fundamental corollaries that mirror Christian teaching regarding our interactions with one another, such as "treat others as you yourself would want to be treated" (because to do so would, in the overwhelming norm, result in the greatest pleasure and least pain, because you would want pleasure and not pain). Common arguments against Hedonism, such that it only concerns itself with physical pleasure and pain, are patently false; when deciding what is moral, all types of pleasure, including mental and emotional, should be considered, and the long-term should also be considered; using drugs, for example, provides some pleasure now, but much more pain in the long term in the form of addiction, health issues, and emotional pain inflicted upon family and friends. However, there are some valid contradictions with "commonsense morality", such as the imperative to save the greater number of people from harm at the expense of those with whom you have the greater emotional tie (most people would reject the idea of sacrificing their perfectly healthy mother in some way to save five strangers; though some Christian scholars would say that's exactly what Jesus taught).
Even outside the bounds of Moral Relativism (which includes Nihilism, Hedonism and Utilitarianism, most of which examine an act by its consequences), moral codes can exist that would result in societies in which we would generally want to live, without relying on imperatives found in any holy text. Kantian moral theory (Immanuel Kant was a devout Christian who sought to logically justify Biblical morality) describes the Categorical Imperative; there are some actions that are categorically moral and others that are categorically immoral. The process for deciding these is not necessarily to consult the Bible, but instead to formulate the imperative, universalize it, and then reason logically to find any contradictions or to reach a conclusion that society as we know it could not exist. For instance, lying is immoral because of a logical contradiction; lying requires meaningful language in order to exist (you have to be able to communicate in order to communicate a falsehood), however if everyone lied, then nobody could tell whether a statement were true or false, and so language would lose its meaning (as the statement could mean what it says or the exact opposite). Murder is immoral for two reasons; first, if everyone murdered at will, humanity would likely die out, and second, there is a logical contradiction that you would not want yourself or your acquaintances to be murdered, but you would want to murder others.